ON CONJUGACY OF p-GONAL AUTOMORPHISMS

• Hidalgo, Ruben A.
• Published : 2012.03.31
• 34 4

Abstract

In 1995 it was proved by Gonz$\acute{a}$lez-Diez that the cyclic group generated by a p-gonal automorphism of a closed Riemann surface of genus at least two is unique up to conjugation in the full group of conformal automorphisms. Later, in 2008, Gromadzki provided a different and shorter proof of the same fact using the Castelnuovo-Severi theorem. In this paper we provide another proof which is shorter and is just a simple use of Sylow's theorem together with the Castelnuovo-Severi theorem. This method permits to obtain that the cyclic group generated by a conformal automorphism of order p of a handlebody with a Kleinian structure and quotient the three-ball is unique up to conjugation in the full group of conformal automorphisms.

Keywords

Riemann surfaces;conformal automorphisms;fixed points

References

1. G. Castelnuovo, Sulle serie algebriche si gruppi di punti appartenenti ad una curve algebraica, Rend. Real Accad. Lincei (5) 15. Memorie scelte, p. 509.
2. H. Farkas and I. Kra, Riemann Surfaces, Second edition. Graduate Texts in Mathematics 71, Springer-Verlag, New York, 1992.
3. G. Gonzalez-Diez, On prime Galois coverings of the Riemann sphere, Ann. Math. Pura Appl. (4) 168 (1995), 1-15. https://doi.org/10.1007/BF01759251
4. G. Gromadzki, On conjugacy of p-gonal automorphisms of Riemann surfaces, Rev. Mat. Complut. 21 (2008), no. 1, 83-87.
5. R. A. Hidalgo, Automorphism groups of Schottky type, Ann. Acad. Sci. Fenn. Math. 30 (2005), no. 1, 183-204.
6. B. Maskit, Kleinian Groups, GMW, Springer-Verlag, 1987.
7. D. Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. (2) 6 (1972), 29-38.
8. A. Wootton, The full automorphism group of a cyclic p-gonal surface, J. Algebra 312 (2007), no. 1, 377-396. https://doi.org/10.1016/j.jalgebra.2007.01.018

Cited by

1. On automorphisms groups of cyclic p-gonal Riemann surfaces vol.57, 2013, https://doi.org/10.1016/j.jsc.2013.05.005
2. On the uniqueness of (p, h)-gonal automorphisms of Riemann surfaces vol.98, pp.6, 2012, https://doi.org/10.1007/s00013-012-0397-8